>> So is this about "getting good at math"? I could see one arguing that>> not much math is involved.>>>> This is arithmetic.>>

> That is not "arithmetic", that is computation, just a part of arithmetic.

So it's not the whole of arithmetic, just part of it.

We know that these same contestants are learning all four algorithmswith that abacus, and are developing the same basic fluency aboutunits, unit conversion, fractions, decimals, as other kids, so FlashAnzan is just a part of arithmetic for them as well. Using an abacusgives them an advantage.

American students taught without it will never be as good atcomputation, in the main. The Americans may be stronger in otherways, every culture makes its investments. The English tend to bestronger than the Americans in vocabulary and awareness of history.[1]

Grade school also includes the concept of prime versus compositenumber. This is where I think grade school level numeracy could domore to introduce higher math concepts and start introducing a firstcomputer language at the same time. I'd want to see Euclid's Methodexplained -- usually bleeped over in today's post US Civil Wartextbooks (not saying it was in them previously either).

Prime vs composite includes "relatively prime" i.e. no factors incommon. gcd(a,b) == 1. This is important for knowing when a fraction(in the sense of a rational number, p/q) is in lowest terms: gcd(p,q)== 1.

Right now, it seems to be college computer science and/or numbertheory that pick up Euclid's Method and extend it. You'll find it inKnuth's TAOCS (The Art of Computer Science) for sure, a verymathematical treatise packed with ideas that are accessible to youngsupple minds as well as creaky old ones.

> And your examples are sport, which is fine. Arithmetic also includes> recognizing the arithmetic in various scenarios and contexts. Knowing when> to add or multiply. Recognizing basic and fundamental patterns like the> commutative and distributive properties. Understanding the nature of> physical quantities and the relationship of the "value" and the "unit" to> them. Knowing how to write and say numbers and use the 13 characters (0-9> period comma minus). Understanding place value. Knowing fractions, both> vulgar and decimal, and recognizing that they are numbers and have their> place on the number line.

We should do more with non-commutative sooner. If you hold a businessenvelop towards you, address facing you and right side up, and dothese two operations: flip away (so lying on its back), flip right (asif axis through your chest, envelope pivots right), the outcome isdifferent than if you pivot right first, then flip it back (axishorizontal).

Given matrices express rotation (I prefer introducing matrices asrotation devices rather than systems of linear equations to be"reduced"), and their multiplication is non-commutative, that wouldmake sense. Plus we have the computer to do the multiplying (withprograms the students wrote, perhaps as teams, perhaps in more thanone language).

More vectors and matrices before college! Is that still arithmetic then?

Spatial geometry needn't have a whole lot of algebra in it at first.I like to stuff data for various polyhedrons into SQL tables.

(1) Main table: one line per polyhedron, gives each an id.

(2) Faces table gives each face going around in a loop, using vectorsas nodes. From the face table you can figure edges as between any twoconsecutive vectors to the same face.

(3) And finally I have the vectors table. 26 vectors happen to workwell for this apparatus, so A-Z.

Might we call this arithmetic? Or does the appearance of 2nd roots(as when figuring lengths) mean we're outside arithmetic all of asudden?

which whole numbers you won't see in Education Mafia schools, but you*will* see in more Asia-influenced Education Yakuza schools (whereI've been teaching, sometimes as a guest teacher, though I did teachhigh school full time for two consecutive years, just a few miles fromthe Statue of Liberty).

Having a unit-volume tetrahedron and a model of 3rd powering thatshows a growing-shrinking regular tetrahedron instead of a cube isstrictly outside Mafia Math (MM).

It's not like the Yakuza schools teach that to the exclusion of thetraditional / conventional unit-volume cube 3rd powering = cubingapproach, just that our students learn that's cultural / ethnic, notuniversal as if the ETs had to believe it.

Indeed, in Martian Math, the way I teach it now [2], the Martians andEarthlings are collaborating on building a dam (joint venture) in adeep canyon. They have different unit measure for concrete, theMartians using a tetrahedron of edges D, the Earthlings / Americansusing a cube of edges R, were R == (1/2) * D.

I wonder if this is algebra already, since R and D are but letters,referring the the Radius and Diameter of a sphere.

>> You really don't appreciate all of this until you actually teach a young> student all of this. We take it for granted. Also, we don't remember the> experience of learning all of this because it occurs before we know enough> to put it in perspective. We might remember the setting, but not the> experience. We remember what it was like to learn algebra but not what it> was like to learn our ABCs or to count or our first exposure to adding and> subtracting. By the time we start to have genuine memories of learning late> in primary school, or sometimes not till middle school, after the basics> have been covered.>

I don't think you should build adult forgetfulness into your theoriesthough it's true for some adults. I have an excellent memory, maybebecause my trajectory was not uniform, meaning it doesn't all blendtogether in retrospect. I remember a misconception I had about thelong division algorithm, in 3rd grade (first form, a British school),that I had to get over. In 2nd grade, we were switching to New Math.We did lots with an abacus with colored beads, but not the Japanesekind with fives.

I remember learning to read and my nose getting in the way at first asI learned to focus.

> Besides its practical importance, arithmetic is about becoming familiar with> numbers, especially real numbers. What could be more important to higher> math? I didn't say "also important" I said "more important"?>

So real numbers are included eh? I would think drawing a boundary atthe rational numbers would be helpful for distinguishing "arithmetic"from "not arithmetic".

Your thinking is always full of surprises because you use your wordsthe way you do, meaning what you mean, without much awareness thatyour specific usage patterns are like finger prints, unique to theindividual, even if there's a family resemblance to other fingers.This is true with everyone of course, but you seem less aware of youruniqueness. In contrast, I have to be very aware of my uniquenessbecause I'm always bringing in important differences.

For example, I have this world map I bring to classrooms that looksreally strange. It's not a Mercator nor even a Snyder Projection(favored by National Geographic). I talk about its history and howexotic it is, and how they won't see it very often around school.This brands me as an outsider, more like an ET. I'm accustomed toplaying that role. I'm invited into classrooms to tell them thingsmost of the teachers have never learned either. That impresses them.

> When you actually teach children, what to teach comes pretty easy, but I> suppose that depends on what kind of teacher you are. I start a lesson with> my son and I know right away if I am missing something. When you have the> teaching bug you are able to be both the student and the teacher at the same> time. This becomes more and more difficult when the class is larger and more> diverse in level. Obviously, it breaks down completely when the class has a> 100+ students, most of them unprepared and uninterested, like we are seeing> in these large required college introductory classes.>> Bob Hansen

I have this intuition that as your son gets older you're going to keeptracking along, with your focus shifting into higher grades. You arefocused on elementary school math right now because that's the currentstate of your nuclear family.

However, I think curriculum writing / designing / assessing needs totake into account the whole picture, i.e. not just time slices.

What are the moving targets and how is what you're learning in 4thgrade going to prove relevant in 12th grade etc.?

Show us the whole map and your choices based on its study. If you just go piecemeal, you might turn down a blind alley.Talk about vectors more, and when you think we should introduce them,if you want to sound more credible as a curriculum assessor.

As we all know, the Education Mafia does not have the power to changefrom within except by lots of things breaking, crises, melodrama etc.

As an ET / Yakuza / Asian [2] I get to star in more of a "Marsattacks" type scenario wherein I go to the parent-age adults (whetherthey have kids or not) and show them how these other adults (theteachers), being Earthlings / Americans, are thereby not especiallyconversant with our alien brand of schooling.

"Rhombic dodecahedron? You've *got* to be kidding me" is a firstresponse, before they're quickly overwhelmed by it's space-fillingrelevance.

Your son is probably not going to learn much about polyhedrons or V +F == E + 2 if going to school in post Civil War Florida. That's justnot in the cards in that state at this time in my view (except maybein the Orlando / Cape Canaveral area?).

However, given the Internet, you have the power to import learningmaterials, including the more Asian-flavored stuff, like SingaporeMath (a moving target). You, the dad, even read the ET stuffdirectly, by tuning in Portland via math-teach, an Asian / Cascadiancapital [4] (with strong alliances to other North American capitals(some of them)).

You may be more influenced than you think by this exposure. If youever catch yourself teaching that 3rd powering could be modeled by atetrahedron, take a moment to praise Portlandia maybe.[5]

Kirby

[1] http://youtu.be/dABo_DCIdpM (I like this guy's ability to doEngish accents but the language is foul, exactly they way kids talk atrecess when the teacher's back is turned, and so more of thatdangerous Youtube stuff (part of what holds American kids back is theraging hypocrisy of the adults, where they act all Puritanical incontradiction to how they behaved as children, which children tend tobe bawdier in their vocabularies than teachers, more fluent in thatrespect (in Portland we celebrate pirates and piracy which helps upour tolerance level for foul language (just read our local weeklies)).